Conical or hemispherical shell composite

7 \includegraphics[max width=\textwidth, alt={}, center]{9c82b387-8e5e-48b9-973d-5337b4e56a66-4_553_630_258_753} The diagram shows a container which consists of a bowl of weight 14 N and a handle of weight 8 N . The bowl of the container is in the form of a uniform hemispherical shell with centre \(O\) and radius 0.3 m . The handle is in the form of a uniform semicircular arc of radius 0.3 m and is freely hinged to the bowl at \(A\) and \(B\), where \(A B\) is a diameter of the bowl.

1. Calculate the distance of the centre of mass of the container from \(O\) for the position indicated in the diagram, where the handle is perpendicular to the rim of the bowl.
2. Show that the distance of the centre of mass of the container from \(O\) when the handle lies on the rim of the bowl is 0.118 m , correct to 3 significant figures. In the case when the handle lies on the rim of the bowl, the container rests in equilibrium with the curved surface of the bowl on a horizontal table.
3. Find the angle which the plane containing the rim of the bowl makes with the horizontal.

6 \includegraphics[max width=\textwidth, alt={}, center]{727412ec-d783-4392-8b84-e7d5435a3f4e-3_424_953_255_596} An object is formed by joining a hemispherical shell of radius 0.2 m and a solid cone with base radius 0.2 m and height \(h \mathrm {~m}\) along their circumferences. The centre of mass, \(G\), of the object is \(d \mathrm {~m}\) from the vertex of the cone on the axis of symmetry of the object. The object rests in equilibrium on a horizontal plane, with the curved surface of the cone in contact with the plane (see diagram). The object is on the point of toppling.

1. Show that \(d = h + \frac { 0.04 } { h }\).
2. It is given that the cone is uniform and of weight 4 N , and that the hemispherical shell is uniform and of weight \(W \mathrm {~N}\). Given also that \(h = 0.8\), find \(W\).

6 \includegraphics[max width=\textwidth, alt={}, center]{add3948c-3b45-4e67-ac84-e2ca935afd64-08_442_953_237_596} An object is formed by joining a hemispherical shell of radius 0.2 m and a solid cone with base radius 0.2 m and height \(h \mathrm {~m}\) along their circumferences. The centre of mass, \(G\), of the object is \(d \mathrm {~m}\) from the vertex of the cone on the axis of symmetry of the object. The object rests in equilibrium on a horizontal plane, with the curved surface of the cone in contact with the plane (see diagram). The object is on the point of toppling.

1. Show that \(d = h + \frac { 0.04 } { h }\).
2. It is given that the cone is uniform and of weight 4 N , and that the hemispherical shell is uniform and of weight \(W \mathrm {~N}\). Given also that \(h = 0.8\), find \(W\).

4. \begin{figure}[h] \end{figure} A thin uniform right hollow cylinder, of radius \(2 a\) and height \(k a\), has a base but no top. A thin uniform hemispherical shell, also of radius \(2 a\), is made of the same material as the cylinder. The hemispherical shell is attached to the end of the cylinder forming a container \(C\). The open circular rim of the cylinder coincides with the rim of the hemispherical shell. The centre of the base of \(C\) is \(O\), as shown in Figure 3.

1. Show that the distance from \(O\) to the centre of mass of \(C\) is $$\frac { \left( k ^ { 2 } + 4 k + 4 \right) } { 2 ( k + 3 ) } a$$ The container is placed with its circular base on a plane which is inclined at \(30 ^ { \circ }\) to the horizontal. The plane is sufficiently rough to prevent \(C\) from sliding. The container is on the point of toppling.
2. Find the value of \(k\).

3. A square ABCD of side 4a is made from thin uniform cardboard. The centre of the square is 0 . A circle with centre 0 and radius \(\frac { 7 a } { 4 }\) is then removed from the square to form a template T, shown shaded in Figure 3.
A right conical shell, with no base, has radius \(\frac { 7 a } { 4 }\) and perpendicular height \(6 a\).
The shell is made of the same thin uniform cardboard as T.
The shell is attached to T so that the circumference of the end of the shell coincides with the circumference of the circle centre 0 , to form the hat H , shown in Figure 4.
[0pt] [The surface area of a right conical shell of radius r and slant height I is \(\pi r l\).]

1. Show that the exact distance of the centre of mass of H from O is $$\frac { 175 \pi a } { ( 63 \pi + 128 ) }$$ A fixed rough plane is inclined to the horizontal at an angle \(\alpha\). The hat H is placed on the plane, with ABCD in contact with the plane, and AB parallel to a line of greatest slope of the plane. The plane is sufficiently rough to prevent the hat from sliding down the plane. Given that the hat is on the point of toppling,
2. find the exact value of \(\tan \alpha\), giving your answer in simplest form.

1. \begin{figure}[h] \end{figure} A hollow toy is formed by joining a uniform right circular conical shell \(C\), with radius \(4 a\) and height \(3 a\), to a uniform hemispherical shell \(H\), with radius \(4 a\). The circular edge of \(C\) coincides with the circular edge of \(H\), as shown in Figure 1. The mass per unit area of \(C\) is \(\lambda\) and the mass per unit area of \(H\) is \(k \lambda\) where \(k\) is a constant.
Given that the centre of mass of the toy is a distance \(4 a\) from the vertex of the cone, find the value of \(k\).

2. A closed container \(C\) consists of a thin uniform hollow hemispherical bowl of radius \(a\), together with a lid. The lid is a thin uniform circular disc, also of radius \(a\). The centre \(O\) of the disc coincides with the centre of the hemispherical bowl. The bowl and its lid are made of the same material.

1. Show that the centre of mass of \(C\) is at a distance \(\frac { 1 } { 3 } a\) from \(O\). The container \(C\) has mass \(M\). A particle of mass \(\frac { 1 } { 2 } M\) is attached to the container at a point \(P\) on the circumference of the lid. The container is then placed with a point of its curved surface in contact with a horizontal plane. The container rests in equilibrium with \(P , O\) and the point of contact in the same vertical plane.
2. Find, to the nearest degree, the angle made by the line \(P O\) with the horizontal.

3. \begin{figure}[h] \end{figure} Figure 2 shows a container in the shape of a uniform right circular conical shell of height 6r. The radius of the open circular face is \(r\). The container is suspended by two vertical strings attached to two points at opposite ends of a diameter of the open circular face. It hangs with the open circular face uppermost and axis vertical. Molten wax is poured into the container. The wax solidifies and adheres to the container, forming a uniform solid right circular cone. The depth of the wax in the container is \(2 r\). The container together with the wax forms a solid \(S\). The mass of the container when empty is \(m\) and the mass of the wax in the container is \(3 m\).

1. Find the distance of the centre of mass of the solid \(S\) from the vertex of the container. One of the strings is now removed and the solid \(S\) hangs freely in equilibrium suspended by the remaining vertical string.
2. Find the size of the angle between the axis of the container and the downward vertical.

4. \begin{figure}[h] \end{figure} A thin uniform right hollow cylinder, of radius \(a\) and height \(4 a\), has a base but no top. A thin uniform hemispherical shell, also of radius \(a\), is made of the same material as the cylinder. The hemispherical shell is attached to the open end of the cylinder forming a container \(C\). The open circular rim of the cylinder coincides with the rim of the hemispherical shell. The centre of the base of \(C\) is \(O\), as shown in Figure 3.

1. Find the distance from \(O\) to the centre of mass of \(C\). The container is placed with its circular base on a plane which is inclined at \(\theta ^ { \circ }\) to the horizontal. The plane is sufficiently rough to prevent \(C\) from sliding. The container is on the point of toppling.
2. Find the value of \(\theta\).

3 \begin{figure}[h] \end{figure} A uniform conical shell has mass 0.2 kg , height 0.3 m and base diameter 0.8 m . A uniform hollow cylinder has mass 0.3 kg , length 0.7 m and diameter 0.8 m . The conical shell is attached to the cylinder, with the circumference of its base coinciding with one end of the cylinder (see Fig. 1).

1. Show that the distance of the centre of mass of the combined object from the vertex of the conical shell is 0.47 m . \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{8e1225a2-cb98-4b71-a4af-0150f093f852-2_497_572_1836_788} \captionsetup{labelformat=empty} \caption{Fig. 2}
   \end{figure} The combined object is freely suspended from its vertex and is held with its axis horizontal. This is achieved by means of a wire attached to a point on the circumference of the base of the conical shell. The wire makes an angle of \(80 ^ { \circ }\) with the slant edge of the conical shell (see Fig. 2).
2. Calculate the tension in the wire.

4 A solid uniform cone has height 8 cm , base radius 5 cm and mass 4 kg . A uniform conical shell has height 10 cm , base radius 5 cm and mass 0.4 kg . The two shapes are joined together so that the circumferences of their circular bases coincide.

1. Find the distance of the centre of mass of the shape from the common circular base. \includegraphics[max width=\textwidth, alt={}, center]{74eaa61a-1507-4cef-8f97-5c1860bdc36a-3_974_1141_484_463} The object is suspended with a string attached to the vertex of the cone and another string attached to the vertex of the conical shell. The object is in equilibrium with the strings vertical and the axis of symmetry of the object horizontal (see diagram).
2. Find the tension in each string.

4 In this question you may use the following facts: as illustrated in Fig. 4.1, the centre of mass, G, of a uniform thin open hemispherical shell is at the mid-point of OA on its axis of symmetry; the surface area of this shell is \(2 \pi r ^ { 2 }\), where \(r\) is the distance OA. \begin{figure}[h] \end{figure} A perspective view and a cross-section of a dog bowl are shown in Fig. 4.2. The bowl is made throughout from thin uniform material. An open hemispherical shell of radius 8 cm is fitted inside an open circular cylinder of radius 8 cm so that they have a common axis of symmetry and the rim of the hemisphere is at one end of the cylinder. The height of the cylinder is \(k \mathrm {~cm}\). The point O is on the axis of symmetry and at the end of the cylinder. \begin{figure}[h] \end{figure} \begin{figure}[h] \end{figure}

1. Show that the centre of mass of the bowl is a distance \(\frac { 64 + k ^ { 2 } } { 16 + 2 k } \mathrm {~cm}\) from O . A version of the bowl for the 'senior dog' has \(k = 12\) and an end to the cylinder, as shown in Fig. 4.3. The end is made from the same material as the original bowl.
2. Show that the centre of mass of this bowl is a distance \(6 \frac { 1 } { 3 } \mathrm {~cm}\) from O . This bowl is placed on a rough slope inclined at \(\theta\) to the horizontal.
3. Assume that the bowl is prevented from sliding and is on the point of toppling. Draw a diagram indicating the position of the centre of mass of the bowl with relevant lengths marked. Calculate the value of \(\theta\).
4. If the bowl is not prevented from sliding, determine whether it will slide when placed on the slope when there is a coefficient of friction between the bowl and the slope of 1.5.

6. \begin{figure}[h] \end{figure} Figure 1 shows a bowl formed by removing from a solid hemisphere of radius \(\frac { 3 } { 2 } r\) a smaller hemisphere of radius \(r\) having the same axis of symmetry and the same plane face.

1. Show that the centre of mass of the bowl is a distance of \(\frac { 195 } { 304 } r\) from its plane face.
   (7 marks)
   The bowl has mass \(M\) and is placed with its curved surface on a smooth horizontal plane. A stud of mass \(\frac { 1 } { 2 } M\) is attached to the outer rim of the bowl. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{8b7133ed-3748-46cb-99d2-570ee33c7393-4_517_729_1318_539} \captionsetup{labelformat=empty} \caption{Fig. 2}
   \end{figure} When the bowl is in equilibrium its plane surface is inclined at an angle \(\alpha\) to the horizontal as shown in Figure 2.
2. Find tan \(\alpha\).
   (6 marks)

3. \begin{figure}[h] \end{figure} A uniform solid hemisphere \(H\) has radius \(2 a\). A solid hemisphere of radius \(a\) is removed from the hemisphere \(H\) to form a bowl. The plane faces of the hemispheres coincide and the centres of the two hemispheres coincide at the point \(O\), as shown in Figure 2. The centre of mass of the bowl is at the point \(G\).

1. Show that \(O G = \frac { 45 a } { 56 }\) Figure 3 below shows a cross-section of the bowl which is resting in equilibrium with a point \(P\) on its curved surface in contact with a rough plane. The plane is inclined to the horizontal at an angle \(\alpha\) and is sufficiently rough to prevent the bowl from slipping. The line \(O G\) is horizontal and the points \(O , G\) and \(P\) lie in a vertical plane which passes through a line of greatest slope of the inclined plane. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{d4fc2ea6-3ffc-42f2-b462-9694adfe2ec1-10_812_1086_1667_493} \captionsetup{labelformat=empty} \caption{Figure 3}
   \end{figure}
2. Find the size of \(\alpha\), giving your answer in degrees to 3 significant figures.

A cylindrical container is open at the top. The curved surface and the circular base of the container are both made from the same thin uniform material. The container has radius \(0.2 \text{ m}\) and height \(0.9 \text{ m}\).

1. Show that the centre of mass of the container is \(0.405 \text{ m}\) from the base. [3]

The container is placed with its base on a rough inclined plane. The container is in equilibrium on the point of slipping down the plane and also on the point of toppling.

1. Find the coefficient of friction between the container and the plane. [3]

\includegraphics{figure_4} An object is composed of a hemispherical shell of radius \(2a\) attached to a closed hollow circular cylinder of height \(h\) and base radius \(a\). The hemispherical shell and the hollow cylinder are made of the same uniform material. The axes of symmetry of the shell and the cylinder coincide. \(AB\) is a diameter of the lower end of the cylinder (see diagram).

1. Find, in terms of \(a\) and \(h\), an expression for the distance of the centre of mass of the object from \(AB\). [4]

The object is placed on a rough plane which is inclined to the horizontal at an angle \(\theta\), where \(\tan \theta = \frac{2}{5}\). The object is in equilibrium with \(AB\) in contact with the plane and lying along a line of greatest slope of the plane.

1. Find the set of possible values of \(h\), in terms of \(a\). [4]

An open container \(C\) is modelled as a thin uniform hollow cylinder of radius \(h\) and height \(h\) with a base but no lid. The centre of the base is \(O\).

1. Show that the distance of the centre of mass of \(C\) from \(O\) is \(\frac{1}{4}h\). [5]

The container is filled with uniform liquid. Given that the mass of the container is \(M\) and the mass of the liquid is \(M\),

1. find the distance of the centre of mass of the filled container from \(O\). [5]

[The centre of mass of a uniform hollow cone of height \(h\) is \(\frac{1}{3}h\) above the base on the line from the centre of the base to the vertex.] \includegraphics{figure_1} A marker for the route of a charity walk consists of a uniform hollow cone fixed on to a uniform solid cylindrical ring, as shown in Figure 1. The hollow cone has base radius \(r\), height \(9h\) and mass \(m\). The solid cylindrical ring has outer radius \(r\), height \(2h\) and mass \(3m\). The marker stands with its base on a horizontal surface.

1. Find, in terms of \(h\), the distance of the centre of mass of the marker from the horizontal surface. [5]

When the marker stands on a plane inclined at arctan \(\frac{1}{12}\) to the horizontal it is on the point of toppling over. The coefficient of friction between the marker and the plane is large enough to be certain that the marker will not slip.

1. Find \(h\) in terms of \(r\). [3]

A box is to be assembled in the shape of the cuboid shown in Fig. 3.1. The lengths are in centimetres. All the faces are made of the same uniform, rigid and thin material. All coordinates refer to the axes shown in this figure. \includegraphics{figure_3.1}

1. The four vertical faces OAED, ABFE, FGCB and CODG are assembled first to make an open box without a base or a top. Write down the coordinates of the centre of mass of this open box. [1]

The base OABC is added to the vertical faces.

1. Write down the \(x\)- and \(y\)-coordinates of the centre of mass of the box now. Show that the \(z\)-coordinate is now 1.875. [5]

The top face FGDE is now added. This is a lid hinged to the rest of the box along the line FG. The lid is open so that it hangs in a vertical plane touching the face FGCB.

1. Show that the coordinates of the centre of mass of the box in this situation are \((10, 2.4, 2.1)\). [6]

[This question is continued on the facing page.] The box, with the lid still touching face FGCB, is now put on a sloping plane with the edge OA horizontal and the base inclined at \(30°\) to the horizontal, as shown in Fig. 3.2. \includegraphics{figure_3.2} The weight of the box is 40 N. A force \(P\) N acts parallel to the plane and is applied to the mid-point of FG at \(90°\) to FG. This force tends to push the box down the plane. The box does not slip and is on the point of toppling about the edge AO.

1. Show that the clockwise moment of the weight of the box about the edge AO is about 0.411 Nm. [4]
2. Calculate the value of \(P\). [2]

The figure show a wine glass consisting of a hemispherical cup of radius \(r\), a cylindrical solid stem of height \(r\) and a circular base of radius \(r\). The cup has mass \(M\) and the stem has mass \(m\). Modelling the cup as a thin, uniform hemispherical shell, the base as a uniform lamina made of the same thin material as the cup, and the stem as a uniform solid cylinder,

1. show that the mass of the circular base is \(\frac{1}{2}M\). [1 mark]

Given that the centre of mass of the glass is at a distance \(\frac{13r}{14}\) from the base along the vertical axis of symmetry,

1. express \(M\) in terms of \(m\). [6 marks]

If the cup is now filled with liquid whose mass is \(2M\),

1. show that the position of the centre of mass rises through a distance \(\frac{13r}{35}\). [6 marks]
2. State an assumption that you have made about the liquid. [1 mark]

\includegraphics{figure_6}